1. STATISTIK PENDIDIKAN EDU5950 SEM1 2013-14
STATISTIK DESKRIPTIF: UKURAN SEBARAN
Rohani Ahmad Tarmizi - EDU5950

2. UKURAN-UKURAN SEBARAN
JULAT
SISIHAN MIN
VARIANS
SISIHAN PIAWAI

3. UKURAN SEBARAN
Setelah mempelajari ukuran kecenderungan memusat UKURAN TAHAP , maka persoalannya seterusnya adalah bagaimanakah skor-skor itu bersebar sama ada tersebar-sebar atau terkumpul-kumpul.
Ini membawa kepada konsep UKURAN SEBARAN IA ITU suatu indeks atau petunjuk sejauh mana skor-skor dalam taburan tersebar.
Ukuran kecenderungan memusat dan ukuran sebaran merupakan petunjuk yang sangat penting untuk data kuantitatif, oleh itu sangat kerap digunakan oleh penyelidik dan dilaporkan dalam sesuatu penulisan.

4. TEKNIK MEMPERIHAL DATA - UKURAN SEBARAN
JULAT - ukuran paling mudah tetapi kasar
SISIHAN MIN - ukuran purata beza mutlak bagi skor-skor daripada min
VARIANS - purata hasil tambah kuasa dua sisihan skor-skor daripada min
SISIHAN PIAWAI - punca kuasa dua bagi purata hasil tambah kuasa dua sisihan skor-skor daripada min

5. UKS - JULAT Julat adalah skor beza antara skor tertinggi dan terendah
Set A:
Set B:

6. JULAT
Ukuran yang paling mudah ia itu dengan menentukan beza antara skor tertinggi dengan terendah.
Skor yang tinggi (SET B = 18) memberi gambaran bahawa ukuran sebarannya lebih besar daripada (SET A = 6).
Dengan itu kita dapat memperkatakan sungguhpun min kedua-dua set adalah sama tetapi sebarannya berbeza.
Ini menggambarkan kompsisi skor-skor dalam taburan tersebut.
Walau bagaimanapun kegunaan julat adalah terlalu terhad oleh kerana ia mengguna dua skor dalam sesuatu set.
Oleh itu, ia hanya diguna untuk mendapat gambaran yang cepat.

7. SISIHAN MIN
Sisihan min merupakan ukuran purata bagi perbezaan skor-skor daripada min.
Untuk mengira sisihan min
L1: Tentukan min bagi taburan
L2: Tentukan sisihan bagi setiap skor daripada min taburan tersebut
L3: Tentukan nilai mutlak bagi setiap nilai sisihan sisihan
L4: Jumlahkan kesemua sisihan-sisihan
L5: Bahagikan jumlah tersebut dengan bilangan skor dalam taburan tersebut

8. UKS - SISIHAN MIN Set A: 21 22 23 24 25 26 27 X 21 22 23 24 25 26 27SM -3 -2 -1 1 2 3

9. UKS - SISIHAN MIN Set B: 15 18 21 24 27 30 33 X 15 18 21 24 27 30 33SM -9 -6 -3 3 6 9

10. VARIANS
Varians ditakrifkan sebagai purata hasil tambah kuasa dua sisihan-sisihan daripada min.
Untuk mengira varians
L1: Tentukan min bagi taburan
L2: Tentukan sisihan bagi setiap skor daripada min taburan tersebut
L3: Tentukan nilai kuasa dua bagi setiap nilai sisihan-sisihan
L4: Jumlahkan kesemua sisihan-sisihan yang telah dikuasakan dua
L5: Bahagikan jumlah tersebut dengan bilangan skor dalam taburan tersebut.

11. UKS - VARIANS X 21 22 23 24 25 26 27 X- -3 -2 -1 1 2 3 (X-)2 9 4 X1 2 3
(X-)2 9 4 X 15 18 21 24 27 30 33
X- -9 -6 -3 3 6 9
(X-)2 81 36

12. SISIHAN PIAWAI
Sisihan piawai pula ditakrif sebagai punca kuasa dua nilai varians
Ini bermakna setelah menentukan varians, anda boleh kirakan sisihan piawai dengan menentukan nilai kuasa dua bagi varians.
Nilai sisihan piawai adalah kecil dan dikatakan dalam unit yang diukur manakala nilai varians adalah besar oleh kerana ia merupakan hasil kuasa dua sisihan-sisihan.
Oleh itu, nilai sishan piawai lebih cekap bagi menggambarkan sebaran

14. PENGIRAAN VARIANS DATA BERKUMPUL
KELAS
FREK.
X TT
(X TT – )2 f(X TT – )2
5-9 2 7
10-14 11 12
15-19 26 17
20-24 22
Min = 962/56 =
= 17.18

16. PENGIRAAN VARIANS DATA BERKUMPUL
KELAS
FREK.
X TT
(X TT – )2 f(X TT – )2
5-9 2 7
10-14 11 12
15-19 26 17
20-24 22
S2 = /56 =
S =
S = 4.00

17. LATIHAN: PENGIRAAN VARIANS DATA BERKUMPUL
KELAS
FREK.
X TT
(X TT – X) F(X TT – X)2
5-9 2 7
10-14 11 12
15-19 26 17
20-24 22
25-29 8 27

18. RINGKASAN
Ukuran kecenderungan memusat dan ukuran sebaran merupakan ukuran yang paling popular digunakan untuk pemerihalan data disamping menyaji data secara jadual/carta atau graf.
UKM menunjukkan tahap (level) manakala UKS menunjukkan kebolehubahan (homogeneity/heterogeneity)
UKM yang paling kerap digunakan adalah min manakala UKS yang disertai adalah sisihan piawai.
Cuba anda beri sebab kenapa min dan sisihan piawai kerap digunakan.

19. TAFSIRAN UKURAN SEBARAN
Ukuran yang besar menunjukkan sebaran/serakan/variasi yang besar.
Ukuran yang besar mennunjukkan skor-skor adalah heterogen (jauh berbeza-beza).
Ukuran yang yang kecil menunjukkan sebaran/serakan/variasi yang kecil
Ukuran yang kecil menunjukkan skor adalah homogen (hampir sama).

20. RINGKASAN
Ukuran kecenderungan memusat dan ukuran sebaran merupakan ukuran yang paling popular digunakan untuk pemerihalan data disamping menyaji data secara jadual/carta atau graf.
UKM menunjukkan tahap (level) manakala UKS menunjukkan kebolehubahan (homogeneity/heterogeneity)
UKM yang paling kerap digunakan adalah min manakala UKS yang disertai adalah sisihan piawai.
Cuba anda beri sebab kenapa min dan sisihan piawai kerap digunakan.

21. Descriptive Statistics
Lets look at the following set of data from two groups of students undergoing two different approaches in learning The mean, the median and the mode for each were as follows
PBL Approach Traditional Approach
Closely alike
Very different
Use this example to review the measures of central tendency. Both sets of data have the same mean, the same median and the same mode. Students clearly see that the data sets are vastly different though. A good lead in for measures of variation.
Mean = 61.5
Median =62
Mode= 67
Mean = 61.5
Median =62
Mode= 67
Nota Tambahan

22. Measures of Variability/Dispersion
Measure or index which convey about the degree to which the scores differ from one another.
Measures that reflect the amount of variation in the scores of a distribution.
Nota Tambahan

23. Measures of Variability/Dispersion
♠ Provides a measure of the dispersion of your data
♠ Measures include:
i) Range – presented as the lowest to the highest values
ii) Variance – the average of squared deviations from mean
iii) Standard deviation – provides a measure of deviation
from mean which is calculated as square root of the variance
♠ Amongst the three measures of dispersion, standard deviation is the most frequently used.
Nota Tambahan

24. Measures of Variability
Range= Maximum value-Minimum value
Range of scores for Set A = = 11
Range of scores for Set B = = 57
The range only uses 2 numbers from a data set, therefore it is only a rough and quick measure.
Explain that if only one number changes, the range can be vastly changed. If a $56 stock dropped to $12 the range would change. Emphasize the need for different symbols for population parameters and sample statistics.
Nota Tambahan

25. Population Variance Population Variance:
The sum of the squares of the deviations, divided by N.
Each deviation is squared to eliminate the negative sign. Students should know how to calculate these formulas for small data sets. For larger data sets they will use calculators or computer software programs.
Nota Tambahan

26. SET A (PBL APPROACH) - Variance
Each deviation is squared to eliminate the negative sign. Students should know how to calculate these formulas for small data sets. For larger data sets they will use calculators or computer software programs.
Sum of squares
188.50
Nota Tambahan

27. SET B (TRADITIONAL APPROACH- Variance
Each deviation is squared to eliminate the negative sign. Students should know how to calculate these formulas for small data sets. For larger data sets they will use calculators or computer software programs.
Sum of squares
Nota Tambahan

28. Population Standard Deviation
Population Standard Deviation The square root of the population variance.
The variance is expressed in “square units” which are meaningless. Using a standard deviation returns the data to its original units.
The population standard deviation for students in the PBL group
is 4.34
Nota Tambahan

29. Population Standard Deviation
Population Standard Deviation The square root of the population variance.
The variance is expressed in “square units” which are meaningless. Using a standard deviation returns the data to its original units.
The population standard deviation for students in the Traditional
group is 17.29
Nota Tambahan

30. Sample Variance (Set A)
To calculate a sample variance divide the sum of squares by n-1.
Samples only contain a portion of the population. This portion is likely to not contain extreme values. Dividing by n-1 gives a higher value than dividing by n and so increases the standard deviation.
Nota Tambahan

31. Sample Variance (Set B)
To calculate a sample variance divide the sum of squares by n-1.
Samples only contain a portion of the population. This portion is likely to not contain extreme values. Dividing by n-1 gives a higher value than dividing by n and so increases the standard deviation.
Nota Tambahan

32. Sample Standard Deviation (Set A)
The sample standard deviation, s is found by taking the square root of the sample variance.
Samples only contain a portion of the population. This portion is likely to not contain extreme values. Dividing by n-1 gives a higher value than dividing by n and so increases the standard deviation.
Nota Tambahan

33. Sample Standard Deviation (Set B)
The sample standard deviation, s is found by taking the square root of the sample variance.
Samples only contain a portion of the population. This portion is likely to not contain extreme values. Dividing by n-1 gives a higher value than dividing by n and so increases the standard deviation.
Nota Tambahan

34. Summary Range= Maximum value-Minimum value Population Variance
Population Standard Deviation
A summary of the formulas
Sample Variance
Sample Standard Deviation
Nota Tambahan

35. (ΣΧ)² ΣΧ² - n s² = n - 1 (96)² 652 - 15 s² = 15 - 1 37.6 s² = 14
Board Demonstration
Calculate range, variance and std dev for the 3 data set
Raw data
♠ Range = 9 – 3
♠ Variance (s²)
Before you can solve for variance,
you need to determine:
n = 15
ΣΧ = 96
ΣΧ² = 652
♠ Std dev (s)
Data set 1:
5 8
9 7
6 8
6 7
7 6
5 3
7 8 4
(ΣΧ)²
ΣΧ² - n
s² =
n - 1
(96)²
652 - 15
s² =
15 - 1
s = √s²
37.6
s = √2.686
s² = 14
s = 1.639 =
Nota Tambahan

36. (ΣΧ)² ΣΧ² - n s² = n - 1 (146)² 1868 - 15 s² = 15 - 1 446.933 s² = 14
Board Demonstration
Calculate range, variance and std dev for the 3 data set
Raw data
♠ Range = 16 – 3
♠ Variance (s²)
Before you can solve for variance,
you need to determine:
n = 15
ΣΧ = 146
ΣΧ² = 1868
♠ Std dev (s)
Data set 2:
15 8
9 7
16 8
16 7
7 6
15 3
7 8 14
(ΣΧ)²
ΣΧ² - n
s² =
n - 1
(146)²
1868 - 15
s² =
15 - 1
s = √s²
s = √31.924
s² = 14
s = 5.65 =
Nota Tambahan

37. …Cont.
Frequency distribution
♠ Range = 45 – 25
= 20
♠ Variance (s²)
Before you can solve for variance,
you need to determine:
n = 71
ΣfΧ = 2,434
ΣfΧ² = 85,810
♠ Std dev (s)
s = √s²
s = √33.357
= 5.776
= 5.78
Data set:
X f fx__
Total
(ΣfΧ)²
ΣfΧ² -
s² = n n
2434² 71
s² = 71
s² = 71 =
Nota Tambahan

38. …Cont. 3. Grouped Frequency distribution ♠ Range Data set:
Not relevant
♠ Variance (s²)
Before you can solve for variance,
you need to determine:
Xmidpt = 25.5, 35.5 and 45.5
n = 71
Σf Xmidpt = 2,370.5
Σf Xmidpt ² = 82,727.75
♠ Std dev (s)
s = √s²
s = √50.466
= 7.104
Data set:
group f
21 –
31 –
41 –
Total
Nota Tambahan

39. To approximate the mean of the data in a frequency distribution, treat each value as if it occurs at the midpoint of its class. Class midpoint = x.
Grouped Data
n =  f
Emphasize that the data consists of 30 values. Occasionally students will try to divide by the number of classes. Always be sure they check the reasonableness of their answer.
Nota Tambahan

40. To approximate the standard deviation of the data in a frequency distribution, use class midpoint = x.
Grouped Data
n =  f
Class f
Midpoint
67- 78 3
72.5
739.84
79- 90 5
84.5
231.04 8
96.5
10.24
81.92 9
108.5
77.44
696.96
120.5
602.5 30
7061.1
Nota Tambahan